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The product of two positive integers is $936$. Find the greater number, if the integers are in the ratio $13:18$.
A.$27$
B.$31$
C.$36$
D.$41$

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Last updated date: 17th Jun 2024
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Answer
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Hint: It is given that there are two positive integers, so consider the two integers $a$ and $b$. After that, apply the condition given and examine which is the greater number.

Complete step-by-step answer:
We are given that there are two integers.
Now let us assume the two integers $a$ and $b$.
We are given that, the product of two positive integers is $936$, we get,
$\Rightarrow$ $ab=936$
Now write it in terms of any one term we get,
$\Rightarrow$ $a=\dfrac{936}{b}$ …………… (1)
Also, we are given that the integers are in the ratio $13:18$.
$\Rightarrow$ $\dfrac{a}{b}=\dfrac{13}{18}$ …………. (2)
Now substituting (1) in (2) we get,
$\Rightarrow$ $\dfrac{\dfrac{936}{b}}{b}=\dfrac{13}{18}$
Simplifying we get,
$\Rightarrow$ $\dfrac{936}{{{b}^{2}}}=\dfrac{13}{18}$
Now cross multiplying we get,
$\Rightarrow$ $13{{b}^{2}}=936\times 18$
Again, simplifying we get,
$\Rightarrow$ ${{b}^{2}}=\dfrac{936\times 18}{13}$
$\Rightarrow$ ${{b}^{2}}=1296$
Now taking square root on both sides we get,
$\Rightarrow$ $b=\pm \sqrt{1296}$
$\Rightarrow$ $b=\pm 36$
In the problem it is given that the integers are positive integers.
So, taking only positive integers,
$\Rightarrow$ $b=36$
Now substituting $b=36$ in (1) we get,
$\Rightarrow$ $a=\dfrac{936}{36}$
Now simplifying we get,
$\Rightarrow$ $a=26$
Now from assumption, we can say that $b$ is greater integer.
So, $36$ is the greater number.
Therefore, the correct answer is option (C).

Additional information:
The word integer originated from the Latin word “Integer” which means whole. It is a special set of whole numbers composed of zero, positive numbers and negative numbers and denoted by the letter Z. Positive numbers are those numbers which are having a plus sign ($+$). Most of the time, positive numbers are represented merely as a whole number without the plus sign ($+$). Every positive number is greater than zero, as well as negative numbers. On a number line, positive numbers are represented to the right of zero.

Note: Integers are the numbers which can be positive, negative or zero. These numbers are used to perform various arithmetic calculations, like addition, subtraction, multiplication and division. Every positive number is greater than zero, as well as negative numbers. In contrast to positive numbers, negative numbers are numbers symbolized with a minus sign ($-$). Negative numbers are represented to the left of zero on a number line.